Final answer:
The question involves calculating the time period money was invested using the compound interest formula. With the given principal of $100, a rate of 7%, and a final balance of $163, the formula is rearranged to solve for the time in years.
Step-by-step explanation:
The student is asking how long a principal of $100 invested at a rate of 7% was invested to reach a balance of $163. This question relates to the concept of compound interest, which is an interest calculation on the principal plus the accumulated interest. To solve this problem, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
Since we know the principal amount (P = $100), the future amount (A = $163), and the annual interest rate (r = 0.07), we can rearrange the formula to solve for t (time in years).
The formula becomes:
t = log(A/P) / (n log(1 + r/n))
Assuming the interest is compounded once a year (n=1), the equation simplifies to:
t = log(163/100) / log(1.07)
Using a calculator to compute the logs, we find the time (t) the money was invested for. Keep in mind that the calculation needs to be adjusted if the compounding frequency is different than once a year.